Subgroup ($H$) information
| Description: | $Q_{16}.A_4$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$a^{3}, cd^{18}, bc, d^{9}, d^{36}, a^{2}d^{12}, d^{54}$
|
| Derived length: | $3$ |
The subgroup is normal, a direct factor, nonabelian, and solvable.
Ambient group ($G$) information
| Description: | $C_9\times Q_{16}.A_4$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable.
Quotient group ($Q$) structure
| Description: | $C_9$ |
| Order: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Automorphism Group: | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-group.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_{12}\times A_4).C_6.C_2^4$ |
| $\operatorname{Aut}(H)$ | $D_8:C_2\times S_4$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| $\operatorname{res}(S)$ | $D_8:C_2\times S_4$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $D_4\times A_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $D_4\times C_9\times A_4$ |